Question 131108
Remember the perimeter of a rectangle is 


{{{P=2W+2L}}}


Since one side is formed from the side of the barn, this means that we can take out one length (or width, it doesn't matter) to get 



{{{P=2W+L}}}



{{{60=2W+L}}} Plug in the given perimeter 60 (since he only has 60 ft of fencing)


{{{60-2W=L}}} Subtract {{{2W}}} from both sides



{{{L=60-2W}}} Rearrange the equation




Now let's introduce another formula. The area of any rectangle is 


{{{A=L*W}}}



{{{A=(60-2W)*W}}} Plug in {{{L=60-2W}}}



{{{A=W*(60-2W)}}} Rearrange the terms



{{{A=60W-2W^2}}} Distribute



{{{A=-2W^2+60W}}} Rearrange the terms



From now on, let's think of {{{A=-2W^2+60W}}} as {{{y=-2x^2+60x}}} where y is the area and x is the width.





Now the equation is in the form of a quadratic which has a vertex that corresponds with the maximum area. So if we find the y-coordinate of the vertex, we can find the max area.






In order to find find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)

To find the axis of symmetry, use this formula:


{{{x=-b/(2a)}}}


From the equation {{{y=-2x^2+60x}}} we can see that a=-2 and b=60


{{{x=(-60)/(2*-2)}}} Plug in b=60 and a=-2



{{{x=(-60)/-4}}} Multiply 2 and -2 to get -4




{{{x=15}}} Reduce



So the axis of symmetry is  {{{x=15}}}



So the x-coordinate of the vertex is {{{x=15}}}. Lets plug this into the equation to find the y-coordinate of the vertex.



Lets evaluate {{{f(15)}}}


{{{f(x)=-2x^2+60x}}} Start with the given polynomial



{{{f(15)=-2(15)^2+60(15)}}} Plug in {{{x=15}}}



{{{f(15)=-2(225)+60(15)}}} Raise 15 to the second power to get 225



{{{f(15)=-450+60(15)}}} Multiply 2 by 225 to get 450



{{{f(15)=-450+900}}} Multiply 60 by 15 to get 900



{{{f(15)=450}}} Now combine like terms



So the vertex is (15,450)

 
This shows us that the max area is then 450 square feet.




So with a width of 15 ft the fence will have a maximum area of 450 square feet




{{{L=60-2(15)}}} Now plug in {{{w=15}}}



{{{L=60-30}}} Multiply



{{{L=30}}} Subtract



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Answer:


So the dimensions of the garden are


width: 15, length: 30


Also, the max area of the garden is 450 square feet.